Basis Definition and 1000 Threads

Homework Statement Prove that the coordinates of a vector v in a vector space Vn are unique with respect to a given basis B={b1,b2,...,bn} Homework Equations The Attempt at a Solution not sure at all what to do with this

If I want to derive the matrix representation for operator Q in the |S1=1/2 ,m1> |S2=1/2 ,m2 > basis, where |Si,mi> are common eigenstates of S2 , Si,z for the ith particle. And I do it in this way: <↑↑|Q|↑↑> <↑↑|Q|↑↓> <↑↓|Q|↓↑> <↑↑|Q|↓↓> <↑↓|Q|↑↑> <↑↓|Q|↑↓> <↑↓|Q|↓↑> <↑↓|Q|↓↓> ...

Homework Statement See attachment. The Attempt at a Solution I already did parts i and ii (correctly, I hope). On part iii I found 2 linearly independent elements to be: t+1, t^2 - 1. However, I don't understand how to show that these form a basis of W. Because W is a subspace of P2, and P2...

Hey guys There are so many of these damn "Find a basis" questions and I can't get any of them because we never directly learned how...or she never showed us in class...my final exam is tomorrow. Here are some examples of questions: http://184.154.165.18/~devilthe/uploads/1323453294.png...

Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach

Let u,v,w\in V a vector space over a field F such that u≠v≠w. If { u , v , w } is a basis for V. Prove that { u+v+w , v+w , w } is also a basis for V. Proof Let u,v,w\in V a vector space over a field F such that u≠v≠w. Let { u , v , w } be a basis for V. Because { u , v , w } its a basis...

Homework Statement For what value(s) of λ is the set of vectors {(λ^2-5, 1, 0), (2, -2, 3), (2, -3, -3)} form a basis of ℝ^3Homework Equations in order for a vector to form a basis it has to span R3 and the set has to be linearly independent.The Attempt at a Solution i tried doing row...

Homework Statement Let S be any non-empty set, F be a field and V={ f : S -> F such that f(x) = 0 } be a vector space over F. Let f[sub k] (x) : S -> F such that f[sub k] (x) = 1 for k=x, otherwise f[sub k] (x) = 0. Prove that the set { f [sub k] } with k from S is a basis for the vector space...

Homework Statement Give expressions for computing the matrix elements Xmn of the matrix X representing the position operator X in the energy basis (using eigenvectors of the Harmiltonian operator) Also told to consider the example of the harmonic oscillator where energy eigenvalues are...

Homework Statement Homework Equations The Attempt at a Solution 2) No clue.

If we're working in R^n and we consider the elements of a basis for R^n to be the column vectors of an nxn invertible matrix B, then what is the relationship between B and the matrix whose row vectors represent elements of the corresponding dual basis for R^n*? My guess, which Wikipedia helped...

I don't wan't a solution I wan't only instructions how to solve this problem: Find a basis for the span: \vec{a_{1}}=(1,\,-1,\,6,\,0),\,\vec{a_{2}}=(3,\,-2,\,1,\,4),\,\vec{a_{3}}=(1,\,-2,\,1,\,-2),\,\vec{a_{4}}=(10,\,1,\,7,\,3)

I’m currently a physics/math major. I work very hard and am proud of my 4.0 GPA. However, as my peers and professors begin to talk about grad school I realize I don’t have a clue what I'm supposed to do. My goal is to go to Penn State for an advanced degree in some type of engineering or...

Homework Statement Assume the inner product is the standard inner product over the complexes. Let W= Spanhttp://img151.imageshack.us/img151/6804/screenshot20111122at332.png Find an orthonormal basis for each of W and Wperp.. The Attempt at a Solution Obviously I need to use Gram-Schmidt...

Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...

Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...

Homework Statement Find the matrix elements of the Hamiltonian in the energy basis for the ISW. Is it diagonal? Do you expect it to be diagonal? Homework Equations H=\frac{p^2}{2m}+V \frac{d}{dt}\langle Q \rangle = \frac{i}{\hbar} \langle[\hat H, \hat Q] \rangle + \langle...

Homework Statement The matrix is: -2 -2 -4 4 -1 1 2 -2 -1 0 -3 0 -4 1 -7 -2 I know the dimensions for the null space are 2 Homework Equations I know that to find the basis for a null space Ax=0, so I row reduced it and I got 1 0 3 0 0 1 5 -2 0 0 0 0 0 0 0 0 The Attempt...

Homework Statement Let W be the plane 3x + 2y − z = 0 in ℝ3. Find a basis for W perpendicularHomework Equations The Attempt at a Solution I thought a basis for this plane could be generated just by letting x=0 and y=1, finding z and then doing the same thing but this time letting x=1 and y=0...

Hi, I seem to remember there is a book by Steven Weinberg that gives the mathematical basis for tensor calculus for relativity, but the name escapes me. Anyone know what I'm talking about?

in particular, i wonder if the trasposition super operator is basis independent or not. We can always write an operator W as \hat{W}=\sum_{i,j} c_{i,j} |i\rangle\langle j| and for the transposed we obtain \hat{W}^T=\sum_{i,j} c_{j,i} |i\rangle\langle j| we obtain a relation true for each...

Homework Statement Let W be the plane 3x + 2y - z = 0 in R3. Find a basis for W^{\perp}Homework Equations N/A The Attempt at a Solution Firstly, I take some arbitrary vector u = \begin{bmatrix}a\\b\\c\end{bmatrix} that is in W^{\perp}. Then I note that W can be rewritten in terms of the...

So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey...

Hello. First, I'd like to apologize because I don't know where to go ask for homework on linear algebra on the forums so if anyone could please let me know, that would be appreciated. Here's the question: Find a basis for the subspace of R^4 spanned by the given vectors Here's the answer...

Homework Statement I need to prove that, <p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'} i.e. find the position operator in the momentum basis p for p'... It's easy to prove that <x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'} (position operator in position basis for x') since I...

Homework Statement Calculate the partial derivatives (∂f/∂x & ∂f/∂y) Homework EquationsThe Attempt at a Solution really confusing me with the use of the summation and power to 3/2. This is my attempt, most definitely wrong but still tried. ∂f/∂x = x + c1*(2*(x-x1))*([( x-x1 )^2 +...

Hello, I am a chemist and have been working on chemical dynamics. Recently I have started working on some many body interactions. Therein I have found some ideas about Fock Space, Fock Matrix, Fock Space Coherences. These are extensively used to provide characteristic information in...

Hello everyone, I am having difficulty understanding the difference between the basis of a subspace A and and the basis of the range of A. My textbook seems to follow the same approach in determining both. So are they essentially the same?

Homework Statement Find a basis for the solution space of the given homogeneous system. x1 x2 x3 x4 1 2 -1 3 | 0 2 2 -1 6 | 0 1 0 0 3 | 0 The Attempt at a Solution When I reduced to reduced row echelon form i get the following matrix...

Homework Statement Find a basis of U, the subspace of P3 U = {p(x) in P3 | p(7) = 0, p(5) = 0}Homework Equations The Attempt at a Solution ax3+bx2+cx+d p(7)=343a+49b+7c+d=0 p(5)=125a+25b+5c+d=0 d=-343a-49b-7c d=-125a-25b-5c ax3+bx2+cx+{(d+d)/2} -->{(d+d)/2}=2d/2=d...

Problem Given a transformation T : P(t) -> (2t + 1)P(t) where P(t) ϵ P3 (a) Show that transformation is linear. (b) Find the image of P(t) = 2 t^2 - 3 t^3 (c) Find the matrix of T relative to the standard basis ε = {1, t, t^2, t^3} (d) Find the matrix of T relative to the basis β1 = {1...

Homework Statement Find the basis of the solution space W \subset \Re^{4} of the system of linear equations 2x_{1} + 1x_{2} + 2x_{3} +3x_{4} =0 _{ } 1x_{1} + 1x_{2} + 3x_{3} = 0 Homework Equations The basis must span W and be independent. The Attempt at a Solution Solving...

Homework Statement Show that if { v_1, ... , v_k} spans V then {T(v_1), ... , T(v_k)} spans T(v) Homework Equations The Attempt at a Solution So we know that every vector in V can be written as a linear combination of v_1,...v_k thus we only need to show that {T(v_1)...

Homework Statement Given the matrix 0 1 0 0 0 1 -3 -7 -5 Find the eigenspaces for the various eigenvalues Prove that there cannot be a basis of R3 consisting entirely of eigenvectors of AHomework Equations The Attempt at a Solution The...

Homework Statement http://en.wikibooks.org/wiki/Linear_Algebra/Vector_Spaces_and_Linear_Systems/Solutions Problem 14 Can answer be (3,1,2)T (2,0,2)T? also, can I reduce the matrix without transpose? thanks Homework Equations The Attempt at a Solution

Homework Statement So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1) (1 1) so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))...

Homework Statement This is from my first-quarter graduate QM course. Part 4 of this problem asks me to compute the unitary operator U which transforms Sn into Sz, where Sn is the spin operator for spin 1/2 quantized along some arbitrary axis n = icos\phisinθ + jsin\phisinθ + zcosθ.Homework...

Homework Statement The matrix A = 1 1 1 1 -1 0 1 0 1 2 3 2 Express null space and row space of A in terms of their basis vectors. 2. The attempt at a solution I have found the null space to be: x3 [1 -2 1 0]^T + x4 [0 -1 0 1]^T. But my problem is how do i write the final answer correctly...

Homework Statement a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W. b) If U and W are subspaces of a finite-dimensional vector space V, show...

Homework Statement Find a basis for (1, a, a^2) (1, b, b^2) (1, c, c^2) Homework Equations The Attempt at a Solution M(1, a, a^2) + N(1, b, b^2) + K(1, c, c^2) = (0, 0, 0) M + N + K = 0 Ma + Nb + Kc = 0 Ma^2 + Nb^2 + Kc^2 = 0 This is as far as I got. I tried monkeying around with these 3...

I am unable to understand as to how the basis for the tangent space is \frac{\partial}{\partial x_{i}}. Can this be proved ,atleast intuitively? Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then \frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle...

Homework Statement Are the following statements true or false? Explain your answers carefully, giving all necessary working. (1) p_{1}(t) = 3 + t^{2} and p_{2}(t) = -1 +5t +7t^{2} form a basis for P_{2} (2) p_{1}(t) = 1 + 2t + t^{2}, p_{2}(t) = -1 + t^{2} and p_{3}(t) = 7 + 5t -6t^{2}...

Homework Statement "In each of the given cases, decide whether the specified elements of the given vector space V (i) are linearly independent, (ii) span V, and (iii) form a basis. Show all reasoning. V is the space of all infinite sequences (a0, a1, a2, ...) of real numbers v1 =...

Homework Statement I am having trouble finding a basis in a given vector space. I understand how to find a basis of Rn, just find linearly independent vectors that span Rn But how would i find a basis of the set of 3x3 symmetric real matrices? Or Find a basis of real polynomials of...

Homework Statement Prove that symmetric and antisymmetric matrices remain symmetric and antisymmetric, respectively, under any orthogonal coordinate transformation (orthogonal change of basis): Directly using the definitions of symmetric and antisymmetric matrices and using the orthogonal...

Homework Statement [PLAIN]http://img193.imageshack.us/img193/3662/unledmcg.png The Attempt at a Solution I rewrote the whole thing in dictionary x_3 = 15 - 8x_1 - 4x_2 x_4 = 7 - 2x_1 - 6x_2 z = 0 + 22x_1 - 12x_2 x_i \geq 0 1\leq i \leq 4 a) So my basis/bases is x...

Homework Statement Is a set of orthogonal basis vectors for a subspace unique? The attempt at a solution I don't know what this means. Can someone please explain? I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the...

i have the first solution y_1(t) = t for (1-t)y'' + ty' - y = 0. I need to get the 2nd linearly independent using Abels theorem. the integration is messy but i have it set up (sorry no latex); y_2 = (t) * integral to t ( 1/s^2 * exp( -integral to t (s(s+1) ds) ) ds. Could anyone...

Homework Statement (In textbook, given a figure, I cannot redraw that figure in this applet, so I shall describe the question in words) I am given a rectangular xy coordinate system determined by the unit basis vectors i and j and an x'y'-coordinate system determined by unit basis...

Homework Statement The four functions v0 = 1; v1 = t; v2 = t^2; v3 = t^3 form a basis for the vector space of polynomials of degree 3. Apply the Gram-Schmidt procedure to find an orthonormal basis with respect to the inner product: < f ; g >= (1/2)\int 1-1 f(t)g(t) dtHomework Equations ui =...